Collective Defined Contribution Plans

Forschung am IVW Köln
Band 5/2016 (Preprint)
Collective Defined Contribution Plans – Backtesting
based on German capital market data 1955 - 2015
Oskar Goecke
ivwKöln
Institut für Versicherungswesen
Forschung am IVW Köln, Band 5/2016 (Preprint)
Oskar Goecke
Forschungsstelle FaRis
Collective Defined Contribution Plans - Backtesting based
on German capital market data 1955 - 2015
Zusammenfassung
Mit dem Pension Schemes Act 2015 wurden im Vereinigten Königreich (UK) sogenannte shared risk
schemes bzw. defined ambition schemes eingeführt. Man kann diese als Kompromiss zwischen den
DC-Plänen (defined contribution schemes) und DB-Plänen (defined benefit schemes) verstehen. Das
Konzept beinhaltet im Besonderen die CDC-Pläne (collective defined contribution), die in Dänemark
und den Niederlanden eine gewisse Verbreitung haben. CDC-Pläne verbinden zwei Ziele. Zum einen
stellen sie aus der Sicht des Arbeitgebers (Trägerunternehmen) reine CD-Pläne dar, da mit der
Zahlung des Beitrags alle Verpflichtungen erfüllt sind. Aus der Sicht des Arbeitnehmers haben CDCPläne den Vorteil einer kollektiven Absicherung; die Versorgungsleistungen sind auf das persönlich
angesparte Versorgungskapital nicht begrenzt. Im folgenden Aufsatz wollen wir die Vorteile von
CDC-Plänen gegenüber reinen CD-Plänen anhand realer Kapitalmarktdaten für den deutschen Markt
für die Jahre 1955 bis 2015 überprüfen. Nach besten Wissen des Verfassers gibt es bisher nur eine
einzige wissenschaftliche Arbeit, nämlich die von Wesbroom et al. aus dem Jahre 2013, die auf der
Grundlage historischer Marktdaten CDC- und DC-Pläne miteinander vergleicht.
Abstract
The Pension Schemes Act of 2015 in the UK introduced shared risk schemes (also called defined
ambition schemes) which are designed as a compromise between defined contribution (DC) and
defined benefit (DB) schemes. The concept of shared risk schemes includes the design of pension
plans with collective benefits, in particular so called collective defined contribution schemes (CDC)
schemes. They are already quite popular in Denmark and The Netherlands. CDC pension plans
combine two objectives: From the perspective of the plan sponsor (the employer) it looks like a DC
plan, since after having paid the defined contribution the sponsor has no further obligations - unlike
DB plans. From the perspective of the employee, a CDC plan offers collective benefits given by
certain types of collective risk sharing mechanism so that she or he is not exposed to the typical risks
of an individual DC plan. Our contribution is to check the advantage of CDC plans over individual DC
plans on the basis of real market data for the German capital market 1955 – 2015. To the best
knowledge of the author, so far only Wesbroom et al. (2013) have presented a comparable analysis
bases on historical market data. Our CDC plan design is taken from (Goecke 2013), which is
appealingly simple to implement in a real world scenario. DC plans with a high equity ratio had been
badly hit by the dotcom crisis of 2000 and by the financial crises of 2008. Our calculations give strong
evidence that the collective component in CDC plans can effectively protect against equity market
crashes.
Keywords
self-financing pension fund; collective defined contribution plans; intergenerational risk transfer;
asset liability management; resilience.
Collective Defined Contribution Plans – Backtesting based on
German capital market data 1955 - 2015
Oskar Goecke
TH Köln (University of Applied Sciences), Institute for Insurance Studies, Claudiusstrasse 1,
D50678 Koeln, Germany
oskar.goecke@th-koeln.de
Abstract
The Pension Schemes Act of 2015 in the UK introduced shared risk schemes (also called
defined ambition schemes) which are designed as a compromise between defined contribution
(DC) and defined benefit (DB) schemes. The concept of shared risk schemes includes the
design of pension plans with collective benefits, in particular so called collective defined
contribution schemes (CDC) schemes. They are already quite popular in Denmark and The
Netherlands. CDC pension plans combine two objectives: From the perspective of the plan
sponsor (the employer) it looks like a DC plan, since after having paid the defined
contribution the sponsor has no further obligations - unlike DB plans. From the perspective of
the employee a CDC plan offers collective benefits given by certain types of collective risk
sharing mechanism so that she or he is not exposed to the typical risks of an individual DC
plan.
Our contribution is to check the advantage of CDC plans over individual DC plans on the
basis of real market data for the German capital market 1955 – 2015. To the best knowledge
of the author, so far only Wesbroom et al. (2013) have presented a comparable analysis bases
on historical market data.
2
Our CDC plan design is taken from (Goecke 2013), which is appealingly simple to implement
in a real world scenario. DC plans with a high equity ratio had been badly hit by the dotcom
crisis of 2000 and by the financial crises of 2008. Our calculations give strong evidence that
the collective component in CDC plans can effectively protect against equity market crashes.
Classification Codes G11, G17, G22
Key Words: self-financing pension fund; collective defined contribution plans;
intergenerational risk transfer; asset liability management; resilience.
Highlights

We describe a collective saving model and test it with real market data for Germany
1955 to 2015.

We show that CDC plans would have systematically outperformed individual saving
plans.

We demonstrate how collective risk sharing arrangements would have behaved during
equity market crashes (2000, 2008).

We show how intergenerational risk transfer improves the risk-return profile.

We measure the return smoothing effect of collective saving.
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1. Introduction
Occupational pension plans are usually categorized into defined benefit (DB-) and defined
contribution (DC-) plans.1 In a DC plan the employer is only obliged to a pay a certain
(defined) contribution into a pension fund and has no obligations with respect to the benefits.
A typical DC plan provides an individual investment account, which is paid off on retirement.
The accrued capital can then used be to purchase an annuity. DB plans typically promise a
pension linked to salary and years of service in the company. Volatile capital markets, a long
lasting low interest rate environment combined with market oriented accounting standards
have led to a situation that employers abandon DB plans and, if at all, replaced them with DC
plans leaving the fundamental pension risks (in particular the investment risk) to the
employees.
Against this background the British Government has recently introduced a new category of
pension plans, called shared risk schemes or defined ambition schemes.2 In this article we
want to explore collective defined contribution (CDC-) schemes, which can be regarded as a
possible design within the shared risk framework. The basic idea behind CDC plans is that
investment risks are shared between the different generations of savers. The vehicle to enable
the risk-sharing mechanism is a collective reserve. These are assets which are not allocated to
the individual accounts of the savers but belong to the savers collectively. If the assets
perform better than expected then the collective reserve will build up. If assets perform worse
than expected (e.g. after a stock market crash) the assets attributed to the collective reserve are
used to stabilize the performance of the individual accounts.
1
2
International Accounting Standards (IAS) 19.7
Pension Schemes Act 2015, section 3.
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We want to demonstrate how a properly designed CDC plan would have performed in the
past compared to individual saving plans. Our backtesting procedure is based on German
capital market data for the time period Jan. 1955 to Sept. 2015.
The discussion in the UK 3 and other countries on reforming private pension systems has
boosted the literature on alternatives to DC and DB plans. There is a long list of science
articles on intergenerational risk sharing in public or private pension systems. A list of more
technical papers on this subject can be found in (Goecke 2013), the reference list of (Turner
2014) is a compilation of more practice-oriented papers and reports on risk sharing
arrangements.
The following literature review is confined to contributions toward the question to what
extent risk sharing pension plans are in some sense better or worse than individual
arrangements. One approach to compare pension plan designs is to apply the utility
methodology. A utility function is used to measure the welfare gains or losses. Probably the
first who applied a utility function approach to intergenerational risk sharing for funded
pension schemes were Gordon/ Varain (1988), whose work has inspired hundreds of
publications. For example, Cui et al. (2005 and 2011) consider DB schemes where benefits
and / or contributions are adjusted. The authors combine generational accounting methods
with fair-value principles to evaluate future contingent cash flows. The intergenerational
transfer helps to smooth consumption during retirement. The main finding of Cui et al. (2011)
is that only hybrid DB schemes with simultaneous adjustment of contributions and benefits
are able to improve welfare. Gollier (2008) also uses a utility function approach to measure
the welfare effect of intergenerational risk transfer. He compares individual DC plans with
CDC plans with a one-off payment at retirement. He determines a first-best intergenerational
3
A broad discussion for the UK can be found in (DWP, 2008), the US- and the international perspective are set
out in (Mitchell, Shea, 2016).
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risk sharing (with no restriction on funding ratio and borrowing) and a second-best
intergenerational risk sharing under more realistic assumptions. Gollier (2008) concludes that
even in the second-best case intergenerational risk transfer is clearly improving welfare.
Gollier (2008) and Cui et al. (2011) cannot be compare directly since Gollier allows for pathdepent asset allocation while Cui et al. assume a constant equity share
In the same spirit as Cui et al. (2005, 2011) is the work of Hoevenaars/ Ponds (2008). They
examine real life DB pension plans with collective components. The primary purpose of the
paper is demonstrate how certain collective components influence future funding ratios.
All studies mentioned above require an obligatory pension scheme. If this requirement is
dropped, the intergenerational transfer is at stake. The new generation will enter the collective
scheme only if the expected utility is not diminished. This problem is analyzed in detail by
Westerhout (2011). He shows that if the transfer between generations is limited to a certain
degree then the new entrants will continue to participate in the collective scheme.
The DWP consultation paper (DWP 2008, pp. 89) reports on some comparative results for DC
and CDC plans based on Monte-Carlo simulations. The evaluating of 5,000 scenarios for 30
years of saving shows that the underlying CDC plan has higher median pension with a smaller
range compared to an individual DC plan. The higher median level results from the fact that
the underlying CDC plan is fully invested in equities while the individual DC plan has a life
cycle component with a gradual switching to bond investment during the last 5 year before
retirement. The smaller rage of outcomes for the CDC plan can be attributed the risk sharing
component. Wesbroom et al. (2013) is so far the only thorough analysis of CDC plans based
on historical market data. These authors contrast the pension outcomes of standard DC plans
with a sample CDC plan on the basis of historic capital market data for the UK (1930 to
2012). In addition to this, Wesbroom et al. (2013) also present results for the prospective
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behaviour over the period 2013 to 2062. In contrast to the DWP Consultation Paper of 2008,
Wesbroom et al. (2013) give a detailed description of the management rules for the the
sample CDC scheme. The control variable in the sample CDC scheme is the funding level. A
funding level in the range between 90% and 110% requires no action. If funding level is out
of this range, then a revaluation or a one-off reduction of benefits is effected. Wesbroom et al.
(2013) calculations show a clear superiority of CDC plans with respect to size and stability of
replacement rates.
Our paper is organized as follows: In Chapter 2 we present a simple model for CDC plans and
describe the return smoothing mechanism. Chapter 3 contains a short description of the
methods we use to compare saving plans. The results of our back testing calculations are
summarized in Chapter 4, we end with a short comment of the results and an outlook (Chapter
5). A description of the database for the German capital market can be found in the Appendix.
2. A simple CDC model for backtesting
2.1. Model description
In Goecke (2013) a simple time continuous model for a collective saving arrangement was
introduced. This serves as the basis for our discrete time CDC model with monthly
adjustments. We consider a pension fund with the following balance sheet – cf. Fig. 1.
Fig. 1. Balance sheet of a CDC pension funds
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We assume that the pension fund is a pure saving vehicle, i.e. it collects saving contributions
and invests the money in the capital market. Each saver has an individual account, which is
liquidated at retirement age. The total of the deposits of all individuals at time t constitute the
liabilities V(t) of the pension fund. The time value of assets A(t) differs from V(t). That makes
the fundamental difference to a mutual fund for which by definition V(t) ≡ A(t). In our CDC
pension fund the assets should properly back the liabilities V(t), i.e. A(t) ≥ V(t). However,
under certain circumstances a temporary deficit could be acceptable or even unavoidable. The
difference R(t) := A(t) - V(t) we call collective reserve. We measure the reserve status by the
logarithm of the funding ratio, (t):= ln(A(t)/V(t)), which turns out to be a more practical ratio
than R(t)/A(t), the usual reserve ratio. We call (t) the log-reserve ratio or simply reserve
ratio; note that R(t)/A(t) ≤ (t) ≤ R(t)/V(t).
To rule out dynamic effects from a growing or shrinking population, we assume that the
payouts from due saving plans are exactly compensated by the total of all saving
contributions. Therefore the temporal development of A(t) only depends on the performance
of the investment.
Finally, we assume that the pension fund is fully self-financing: There is no external source of
income for the fund other than contributions and investment returns. There is also no external
entity who can withdraw money from the pension fund.
Asset-Liability-Management strategies
We assume that there is a person who manages the pension fund altruistically, i.e. she or he is
acting solely for the benefit of the savers. The pension manager can influence the fund in two
directions: Firstly, she or he can determine the asset allocation (asset management), and
secondly, can decide to what extend assets are attributed to V(t) or to R(t) (liability
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management). More specifically, we assume that at time t the pension manager determines the
risk exposure (t) of the assets for the next time interval [t, t + ]. A risk exposure of (t) = 0
would correspond to a risk free investment, say money market instruments with term = .
The liability management is performed in such a way that at the beginning of every time
interval [t, t + ] the pension manager declares an interest rate on the individual accounts. If
(t) notes the declared (compound) interest rate, then V(t + ) = V(t) exp( (t)). Following
the wording used for with-profit life insurance contracts we call (t) the profit participation
or bonus declaration. However, we should keep in mind that the pension fund does not
provide any interest rate guaranties and that we allow for negative values for (t).
Asset-Liability-Management (ALM) is the simultaneous decision on (t) and  (t) on the basis
of all information up to time t. If  (r)(t):= ln(A(t+)/A(t)) /  denotes the realized rate of
return , then
(t  )  (t)   (r ) (t) (t) .
(1)
Eq. (1) can be regarded as the fundamental ALM-equation of a CDC plan. The actual return
(r)(t) and the profit participation (t) are not necessarily identical. But any difference
between the two must be absorbed by the reserve. An increasing reserve is clearly in the
interest of future generations, while a reduction of the reserve is advantageous especially for
savers whose plans mature in near future. In the case of pure defined contribution (DC-) with
no risk sharing mechanism we have (t)  (r)(t). Then, in view of Eq. (1), (t) will be
constant; actually a non-zero reserve makes no sense for a DC plans. Eq. (1) also indicates the
welfare effect of CDC plans: The individual saving accounts can be protected against capital
market risks - at least to a certain degree.
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Goecke (2013) discusses a simple (linear) ALM-model based on a Black- Scholes-type capital
market model. There  (t) =  (t) , where  (t) is the equity ratio at time t and  is the
constant volatility of the underlying geometric Brownian motion.
The proposed ALM- strategy is characterized by three essentials:
1. The pension manager seeks to keep  (t) close to a time independent strategic reserve
ratio s > 0 . If  (t) =s then the pension manager chooses  (t) =s we; we call s
the strategic risk exposure.
2. The pension manager pursuits a “fair” profit participation. This means that in general
the profit participation follows the expected rate of return, which in turn depends on
the chosen risk exposure.
3. In case of an imbalance (i.e. (t) ≠s) the pension manager takes measures to restore
equilibrium by adjusting profit participation and risk exposure. More specifically
Asset management rule:
 (t )   s  a   (t )  s 
(2)
Liability management rule:  (t )   ( e ) (t )     (t )   s  ,
(3)
where  (e)(t) is the expected return on investment, depending on  (t). Constants a and
 are adjustment parameters that govern the speed of resilience.
Supposing we have always estimated favourably with respect to the asset returns, i.e.  (e)(t) =
 (r)(t), then combining Eq. (1) and Eq. (3) we get
 (t   )   s  (1   )   (t )   s  .
(4)
This means that the liability management rule Eq. (3) ensures that the reserve gap ((t) - s) is
reduced with the constant rate, i.e. we have an exponential damping with factor . For the
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continuous time model one can show that ( (t))t ≥0 and ( (t))t ≥0 are mean reverting processes
with  (t) ≥ 0 and  (t) ≥ s - s /a provided a > 0.4
To apply the above model to real capital market data we have to make a few rather obvious
adjustments. Similar to the Black-Scholes model we assume that our pension fund can invest
into two assets, an equity fund (DAX-portfolio) and a fixed income fund (REXP-portfolio).
DAX and REXP are performance indices mapping a broadly invested German blue chip stock
and a portfolio of German government bonds, resp.5 Our backtesting is performed on a
month-on-month basis, i.e.  = 1/12; starting 01.01.1955. At the beginning of each month, the
pension manager determines the equity share  (t)  [0, 1] for the following month. We
assume a one-to-one correspondence between the risk exposure and equity ratio:  (t) = DAX
 (t), where DAX is the time independent risk exposure of a 100% DAX-investment. To apply
the asset management rule in Eq. (2) we need a suitable calibration for the strategic risk
exposure s and the adjustment parameter a. To apply the liability management rule in Eq. (3)
to real data, we have to determine the expected return on assets  (e)(t) as a function of the risk
exposure  (t). We define the estimated return by
 (e) (t ) : m (t ) 
 (t )
ERP  12  2 (t ) ,
 DAX
(5)
where m(t) is the (historic) 1-month money market rate, observed at time t for an investment
periode [t, t+].6 In Eq (5) ERP denotes the equity risk premium, the extra return for a 100%
stock portfolio. One should note that a REXP-investment is not really risk free since,
depending on the duration, volatile interest rates induce price gains and losses. That is the
reason why in Eq. (4) we take money market rates as a proxy for a risk free investment.
4
cf. (Goecke, 2013), Proposition A.2
For details see Appendix.
6
The data base is explained in the Appendix.
5
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2.2. Calibration and CDC-variants
Generally spoken a capital funded pension system should enable a fair participation of the
savers in the production factor capital, whereas pay-as-you go systems are designed to enable
a fair participation of the retirees in the production factor labor. An easy access to the
production factor capital is to buy shares of a wide range of companies. An investment into
corporate bonds only indirectly secures a share in production factor capital since part of the
profit generated by the company, namely a risk premium, goes to the shareholders. Similar
considerations apply to an investment into government bonds. Thus - at least in theory - only
a 100% real investment would insure a full participation in the production factor capital. But a
100% equity investment means an unacceptable risk for most savers. The idea of CDC plans
is to take a high risk exposure on the investment side and to distribute the risk between the
saver generations.
Another important feature of CDC plans is that they are self-financing. A CDC plan can only
redistribute to the savers what the asset side has earned. In the long run a CDC plan cannot
disassociate from fundamental capital market trends. Therefore, a sustainable ALM-strategy
must automatically adapt to capital market trends, i.e. a CDC plan needs a “breathing” ALMstrategy. With respect to our model, this means that we should not use time independent
parameters, which are calibrated with historic capital market data. However, in Eq. (4) we do
not comply with this rule since we take ERP = 5% and DAX = 0.2. 7 All other parameters (see
Table 1 below) are calibrated by direct calculations (cf. Goecke 2013) or Monte Carlo
simulation (cf. Goecke 2012). Actually, we could make Eq. (4) “breathing” by replacing the
7
In view of literature on calculation and estimation of the ERP a value of 5% seems to be a reasonable
estimation – cf. (Stehle/ Schmidt 2015), (Damodaran 2015), (Dimson et al. 2011). The annualized volatility of a
DAX-investment between 01.1955 and 09.2015 was 19.69%.
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time independent values DAX and ERP by moving averages of observed market values. Then
the ALM-rules would automatically adjust to structural capital market shifts. In other words,
the ALM-rules (2) and (3) ensures that the pension system is resilient against capital market
shocks and structural shifts.
Overview of the backtesting model and calibration
Backtesting period:
Input data:
Parameters:
1 Jan. 1955 (t = 0) – 30 Sept. 2015 (t = T:= 60.75 years = 729 months)
DAX, REXP and money market interest rates   (t ) 
 s:
strategic (log-) reserve ratio
 s: strategic equity share (risk exposure)
:
adjustment (speed) parameter for profit participation
a:
adjustment (speed) parameter for asset allocation
: (log-) reserve ratio at time t = 0
ALM-rules:
 (t )   s  a   (t )   s  , 0   (t )   DAX
Asset allocation
 ( e ) (t )  m (t )  0.25  (t )  12  2 (t )
Expected return on assets
 (t )   ( e ) (t )     (t )   s 
Profit participation
The following Table 1 gives an overview of the variants of CDC plans. We take CDC1 as our
reference model. CDC2 - CDC9 are variants of CDC1 to illustrate the effect of different
calibration of the ALM-parameters. Deviations from the reference calibration of CDC1 are
given in bold figures For all variants we have calculated some key figures with respect to the
(log-) reserve ratio, the equity ratio and the profit participation:
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CDC1 CDC2 CDC3 CDC4 CDC5 CDC6 CDC7 CDC8 CDC9
0
 s
 s

a
0.2
0.0
0.8
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.04
0.16
0.3
0.3
0.3
0.1
0.5
0.3
0.3
0.3
0.3
0.6
0.6
0.6
0.6
0.6
0.1
1.4
0.6
0.6
23.58
29.25
28.19
24.69
-7.73
1.34
3.60
-4.82
Reserve ratio in %
Mean 26.74 26.04 29.77 29.78 25.27
-0.08
-0.08
-0.08
-4.76
2.79
Min
115.91
112.63
129.65
127.46
104.02
Max
Std. dev. 19.24 19.12 22.10 23.50 16.20
last value 24.45 24.45 24.45 31.70 20.42
90.11 118.22 108.99 117.83
15.49
19.79
15.60
21.63
28.35
26.32
26.43
25.04
Equity ratio in %
Mean
Min
Max
Std. dev.
56.86
55.28
60.97
59.06
56.78
51.78
57.41
39.25
71.04
0
0
0
0
0
36.14
0
0
5.53
100
100
100
100
100
85.06
100
100
100
30.69
31.94
31.40
33.39
29.35
7.74
39.62
30.64
27.77
Profit participation (rates per month) in %
Mean
Min
Max
Std. dev.
0.71
0.69
0.80
0.63
0.76
0.63
0.76
0.70
0.69
-0.12
-0.26
-0.12
0.07
-0.31
-0.34
-0.04
0.07
-0.32
3.09
3.01
3.44
1.59
4.20
2.43
3.15
2.92
3.14
0.50
0.50
0.57
0.26
0.68
0.37
0.52
0.42
0.55
Table 1: Overview of CDC-variants,

CDC2 and CDC 3 show the effect of different initial reserve ratios. CDC2 will be of
special interest since this calibration represents a cold start of the model. Since the
ALM rules require the building up of reserves in the initial phase, the first saver
generation will systematically subsidize the reserve.

CDC4 and CDC 5 show the effect of varying parameter . A small  will result in a
more stable profit participation (cf. standard deviation) but also increases the
probability of a negative reserve.

CDC6 and CDC7 show the effect of varying parameter a. If a is small then the asset
allocation will only slowly adjust to changing capital market data. This might result in
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higher risk of underfunding but could also result in missing the chance of higher
returns (if risk exposure is too low).

CDC8 and CDC9 show the effect of different strategic risk exposures s. CDC8 (s =
0.04) represents a target equity ratio of only 20%, which results in a less volatile
reserve ratio and also in a more stable, but lower profit participation. In CDC9 (s =
0.16) the pension manager wants to realize an equity ratio of 80%. In the long run this
allows for a higher profit participation but also jeopardizes the funding ratio.
However, it turns out that the back testing results for these two variants do not differ
too much from each other. This is due to the fact that the average equity ratio for
CDC8 is about 40% (twice the target ratio) and for CDC9 about 70%.
Our aim is to compare CDC plans with individual savings plans. However, we start with an
illustration of how the CDC-model works (cf. Fig. 2a-c). For the sake of clarity, we exhibit
only the first 10 years for the CDC-variants 1,4,5,6 and 7. The effect of the speed parameter a
becomes clear in Fig. 2b by comparing CDC6 (a = 0.1) and CDC7 (a = 1.4). In CDC6 the
equity ratio is only gradually adjusted while CDC7 very much resembles a stop-and-go asset
strategy. Comparing CDC4 ( = 0.1) and CDC5(= 0.5) reveals that a timely adjustment of
profit participation helps to keep the reserve level high (cf. solid blue line in Fig. 2a).
10.02.2016 18:31
15
Fig. 2a-c. Effect of the parameters  and a on reserve ratio (Fig. 2a: top), equity ratio (Fig.
2b: middle) and profit participation (Fig. 2c: bottom)
All CDC variants except for CDC2 and CDC3 start with the initial reserve ratio of  0 = 0.2;
this corresponds to a funding ratio of 122%. This means, that the first generation of savers has
inherited assets that can use used as risk buffer against capital market volatility. The question
arises of what happens if no initial reserve exists. Actually, the problem comes up whenever a
new generation of savers enters a CDC plan. They will ask themselves whether an individual
saving plan would be more favorable than entering an underfunded CDC plan.
We will discuss this issue in more detail in section 3.4. One aspect of this question is
illustrated in Fig. 3 – see below. Here we compare CDC1 ( 0 = 0.2), CDC2 ( 0 = 0) and
10.02.2016 18:31
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CDC3 ( 0 = 0.8) with respect to the reserve ratio. It turns out that after a transitional period of
about 60 month the initial reserve ratio plays no further role. If we look at long term saving
plans, we see that the initial reserve ratio has almost no influence on the final performance (cf.
Fig. 3 below).
Fig. 3. Reserve ratio and profit participation during the first 10 years for different initial
reserve ratios
3. Results
3.1. Risk measures for long term saving plans
Before we present the results, we have to explain how to evaluate the different saving
schemes. Our point of view is that of an employee who sets aside money from her or his
regular salary to build up a pension capital. Upon retirement the pension capital is withdrawn.
We assume that she or he has the choice between
10.02.2016 18:31
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
an individual saving plan where the saving rates are invested into a DAX Portfolio
(DAX plan)

an individual saving plan where the saving rates are invested into a REXP Portfolio
(REXP plan)

an individual saving plan where the saving rates are invested in a money market
account (MM plan)

a CDC plan which is managed as stated above. We analyze the reference CDC plan
(CDC1) and some variants (CDC2 - CDC9).
We neglect any cost components (acquisition cost, administration charge), keeping in mind
that a collective plan might incur fewer administrative costs due to economies of scale
resulting from the collective asset management.8
Figure 4 shows the performance of the different saving vehicles. Except for the first 3 years
investment into a DAX portfolio was always better than investing into a REXP portfolio which
in turn was much better than a money market investment. One should note that in Fig. 4 we
have a log-scale; the final value for DAX (123.72) is roughly factor 3 higher than final REXP
value (44.84) which in turn is about 3 times the value for MM plans (15.41).
Fig. 4. Performance (log scale) of a 1€-investment in 1 Jan. 1955 up to 30 Sept. 2015 in
different saving vehicles: equity (DAX), government bonds (REXP), one month money (MM),
CDC pension fund with initial reserve ratio of 20% (CDC1) or 0% (CDC2)
8
(DWP 2008, p. 91) reports that a collective DC schemes might incur expenses of 0.5 per cent less than for
individual DC plans.
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Fig. 4 also illustrates the risk of a lump sum investment into stocks. There is a twofold timing
risk, the risk of choosing the wrong time to start the investment and the risk of choosing the
wrong time to withdraw the money. A few months starting or ending earlier or later can make
a tremendous difference!
We restrict ourselves to the analysis of long term saving plans with constant saving rates.
Clearly we have to evaluate the return and the risk. We will measure the return as the
annualized rate of return. There is a long list of possible risk indicators measuring the
investment risk – cf. (Bacon, 2012). However, none of these is specially tailored to long term
saving plans. Our choice of risk indicators is motivated heuristically and therefore the choice
is in no way compelling. We use two sets of risk indicators. The first contains risk indicators
that measure the degree of uncertainty of the wealth at maturity. The second set comprises
risk indicators that measure the degree of unsteadiness of the saving process.
3.2. Risk return profiles with respect to the final capital
When evaluating the series of saving plans, we first look from the perspective of savers who
are solely interested in her/ his wealth at maturity. This group of savers does not bother how
their savings evolves - intermediate ups and downs do not cause any stress.
We evaluate 10-, 20-, 30- and 40-year saving plans with respect to the (annualized) yield at
maturity. If we interpret the observed yield at maturity as the realization of a random variable,
then the standard deviation is a natural risk indicator. The minimum and maximum of
observed yields also measure aspects of the risk. As pointed out, the investment into a stock
portfolio means a twofold timing risk. In our context, the timing risk is more or less the risk of
belonging to the wrong generation since the saving period coincides with the working life.
To illustrate this aspect we look at all possible 40-years saving plans.9 Each 40-year period
represents a generation of employees. From a macro economic viewpoint a capital funded
pension system should ensure a fair participation in the production factor capital. In view of
Fig. 5 (see below) it is obvious that the generation of savers who retired in April of 2003 will
definitely not feel fairly treated. To measure the risk of belonging to the wrong generation we
introduce a risk indicator we call intergenerational imbalance. We calculate this as the
9
Between Jan. 1995 and Sept. 2015 there are 250 generations of savers entering work life at age 25 and retiring
at 65.
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maximum difference between the rates of return between “neighboring” generations. To be
more precise, we define
intergenerational imbalance:= max  yt  ys : t  s  1, 0  s, t  T  ,
where yt is the yield at maturity of a saving plan maturing at time t. An intergenerational
imbalance of zero means that all generations have the same provision capital at maturity. The
idea behind this risk indicator is twofold. Firstly, we think that within 1 year period the
“value” of a lifelong participation in production factor capital should not change too much.
Secondly, we believe that in the long run the fundamental economic data can change.
Therefore the value of participation for one generation can be substantially different from that
of a generation born 10 years or more later. If for example, we observed that the yield at
maturity steadily decreases over time, then the standard deviation of these values would
indicate a high risk. However, the steady decline could also be a consequence of a steady
decline of economic growth and it is only fair that pensions are affected. Thus, in this case the
standard deviation would be misleading. In contrast, in this situation the risk parameter
intergenerational imbalance would rightly indicate a low risk.
Fig. 5. Final wealth of 40-years saving plans maturing between 31 Dec 1994 and 30 Sept.
2015 for a monthly rate of 100€.
The following table is based on the evaluation of all 10-, 20-, 30- and 40-year saving plans
with in the time span Jan. 1955 to Sept. 2015.
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1-MM
Minimum
Maximum
Mean
Median
Standard deviation
Intergenerational imbalance
REXP
DAX
CDC1
10y saving plans (610 samples)
CDC2
0.78%
7.46%
3.88%
9.32%
-5.91%
23.09%
5.55%
14.26%
5.55%
14.26%
4.99%
5.59%
1.80%
1.19%
6.69%
6.63%
1.37%
2.49%
7.79%
6.82%
5.71%
17.66%
8.83%
8.39%
2.32%
2.64%
8.77%
8.35%
2.27%
2.64%
20y saving plans (490 samples)
Minimum
Maximum
Mean
Median
Standard deviation
Intergenerational imbalance
1.73%
6.83%
4.68%
8.37%
2.08%
16.70%
6.66%
12.04%
6.65%
12.04%
5.29%
5.73%
1.45%
0.54%
6.96%
7.15%
0.91%
1.09%
8.20%
7.79%
3.43%
8.29%
8.94%
9.28%
1.36%
1.26%
8.92%
9.28%
1.37%
1.26%
30y saving plans (370 samples)
Minimum
Maximum
Mean
Median
Standard deviation
Intergenerational imbalance
2.75%
6.67%
5.49%
8.04%
6.21%
13.61%
8.17%
10.76%
7.98%
10.76%
5.37%
5.81%
1.11%
0.31%
7.07%
7.28%
0.63%
0.59%
8.98%
8.59%
1.61%
4.53%
9.25%
9.06%
0.70%
0.66%
9.23%
9.02%
0.71%
0.68%
40y saving plans (250 samples)
Minimum
Maximum
Mean
Median
Standard deviation
Intergenerational imbalance
3.61%
6.23%
6.07%
7.67%
6.32%
11.04%
8.75%
9.81%
8.68%
9.81%
5.27%
5.40%
0.76%
0.23%
7.03%
7.22%
0.41%
0.40%
8.77%
8.69%
0.91%
3.19%
9.19%
9.19%
0.31%
0.38%
9.17%
9.19%
0.32%
0.39%
Table 2: Evaluation the yield at maturity of 10-, 20-, 30- and 40-year saving plans.
It is worth noting that the intergenerational imbalance is not proportional to the standard
deviation. In particular, MM plans are more risky than REXP plans with respect to standard
deviation but are less risky with respect to intergenerational imbalance. For all durations the
risk of CDC plans is considerably lower than that of DAX plans. For durations of 30 and 40
years the risk of CDC plans is roughly at the level of REXP plans. For shorter durations the
standard deviation of REXP plans is slightly lower than that of CDC plans. It should be noted
that in Table 2 the difference between CDC1 ( 0 = 0.2) and CDC2 ( 0 = 0) can be neglected.
10.02.2016 18:31
21
That means that the generation of savers who entered the CDC scheme realized nearly
identical pension wealth at maturity.
3.3. Path dependent risk measures for saving plans
Compared with an employee who pays contributions and then patiently awaits retirement, in
this section, we take the view of an employee who permanently checks her or his pension
wealth. It is just a matter of transparency that at least once a year a pension fund should
inform its clients about the performance. This is also demanded by consumer organizations.
The rational behind this is that the employee should have the opportunity to react if things
evolve differently from her or his expectations. From this perspective we need path-dependent
risk measures to evaluate the risk of a pension arrangement. Again, there is no single
undisputed path dependent risk measure. We take a rather pragmatic selection of risk
indicators:

Path volatility: For each investment vehicle we consider the time series of the monthly
(log-) returns. The path volatility is then the annualized standard deviation of this time
series. Note that a steady decrease of the returns (e.g. due to declining interest rates)
will indicate a risk with respect to path volatility.

Number of months with negative returns

Maximum Drawdown (MDD): If Sk denotes the accrued capital after k months of
 S  Sk d

:1  k  k  d  n  . MDD is the maximum
saving, then MDD : max  k
Sk


relative loss during the saving time. MDD is a common risk indicator for investment
strategies. However, it is mostly used for lump sum investments and measured as
absolute drawdown; we prefer the relative drawdown.

Maximum Recovery Time (MRT): MRT : max  k  l : Sk  Sk d for all d  1, ..., l  k  . MRT
is the maximal time it takes to recover from losses after a prior peak. It is clear that
MRT = 0 if and only if MDD = 0.
Path volatility and number of negative returns are path dependent risk measure that only
depend on the monthly returns of the underlying vehicle and give the same value for lump
sum investments and for saving plans with constant rates.
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MDD and MRT can be defined for wide range of continuous time stochastic processes.10 In
practice, MDD and MRT are usually applied to lump sum investments.11 We think that MDD
and MRT are good indicators of the savers frustration. We illustrate both concepts in Fig. 6.
The example given is the worst case with respect to MDD among all 30-year DAX plans. A
saver of this generation (starting April 1973) lost 67.8% of the wealth ( 230,000€) within the
last 3 years. In this case the MRT is 40 months. If this generation invested in a DAX plan it
had to bear the full impact of the dotcom crisis. If this generation had invested in a REXP plan
it would not have been affected by the dotcom crises but would have ended up not much
better. An investment into a CDC plan would have definitely been the better choice!
Fig. 6 Performance of a single 30-year saving plan (01.04.1973 - 31.03.2003, monthly rate =
100€). MDD and MRT are indicated for the DAX plan.
We calculate the mentioned path dependent risk indicator for all 370 samples of 30-year
saving plans – cf. Fig. 7a-d below. We notice that the volatility of the DAX has increased
gradually from about 17% to 22%. The REXP-volatility fluctuates around 3.5%. The path
volatility of money market saving (MM-) plans is more or less constant at 0.7%. The average
path volatility of a CDC1 plan is 1.5%. It is interesting to note that the number of months with
negative returns does not differ so much between DAX- and REXP plans (cf. Fig. 7b, chart top
10
(Mahmoud 2015) presents a mathematical framework for continuous time path-dependent risk measures.
MDD for geometric Brownian motion is studied in (Goldberg, Mahmoud 2014), (Zhang et al. 2013).
11
(Bacon 2012) pp. 97ff.
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right). Though the ALM rule (3) allows for negative profit participation for CDC plans, the
historic backtesting shows that negative returns were extremely rare events for CDC plans.
With respect to the maximum drawdown (cf. Fig. 7c, chart bottom left) MM- and CDC1 plans
had been risk free. None of the 370 generations of savers had to suffer from a single
drawdown. In other words, every single contribution increased the pension wealth. The
situation for DAX plans is totally different: On average an employee with an individual DC
plan fully invested in German equities would have suffered an intermediate loss of more than
50%.
The MRT for 30-year saving plans for a pure DAX-investment ranges between 61 and 147
months with an average of 91 months (cf. Fig. 7d, chart bottom right). Thus every long term
saver in a DAX plan experienced a time span of at least 61 months during which she or he
continuously paid in without having one cent more. The two indicators MDD and MLT
apparently measure different aspects of risk and might help to explain why savers very often
terminate their saving plans prematurely.
Fig. 7a-d. Path dependent risk indicators of 30-year saving plans, comparison between
individual plans (DAX-, REXP-, 1-MM plans) and collective plans (CDC1). The calculations
are based on 370 saving plans maturing between 31.12.1984 and 30.09.2015.
Fig. 7a-d show that with respect to all path-depending risk indicators the CDC plan ranges at
the level of money market instruments. From that point of view, a CDC plan is an extremely
10.02.2016 18:31
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safe investment. It turns out that the initial reserve ratio has alomost no effect on the risk
parameters – cf. Table 3.
1-MM
REXP
DAX
CDC1
CDC2
Evaluation of 30y saving plan (370 samples)
Path volatility:
Minimum
Maximum
Mean
Months neg. returns:
Minimum
Maximum
Mean
Max. drawdown:
Minimum
Maximum
Mean
Max. recovery time:
Minimum
Maximum
Mean
0.64%
0.72%
0.70%
3.23%
3.93%
3.66%
16.69%
22.46%
19.52%
1.14%
1.92%
1.53%
1.14%
1.87%
1.52%
0
7
0.1
59
109
90.3
141
161
148.5
0
10
3.5
0
10
3.6
0
0
0
3.18%
5.78%
4.30%
32.63%
67.75%
52.93%
0
0.11%
0.02%
0
0.11%
0.02%
0
0
0
10
16
12.8
27
85
57.2
0
4
0.7
0
4
0.7
Table 3: Path dependent risk indicators for 30-year saving plan – cf. Fig. 7a-d
To visualize the dependency between risk and return, we sketch the relation between the four
path dependent risk indicators (average over all 370 saving plans) and the return, measured as
the average yield at maturity – see Fig. 8a-d. We have included calculations for all variants of
CDC plans, for MM plans and for DAX/ REXP mixed portfolios (0%, 10%, …, 100% DAXshare). It turns out that all variants of CDC plans clearly outperform all variants of individual
saving plans. It should be emphasized that the initial reserve endowment has barely any on the
risk return profile - cf. CDC1, CDC2 and CDC3 in Fig. 8a-d.
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Fig. 8a-d. Risk return-profile of 30-years saving plans: Mean value of annualized yield at
maturity versus four risk indicators: standard deviation (Fig. 8a, top left), intergenerational
imbalance (Fig. 8b, top right), path volatility (Fig. 8c, bottom left) and maximum drawdown
(Fig. 8d, bottom right). We compare individual saving plans (DAX - REXP mixed portfolios,
MM plan) and with CDC-variants (CDC1-CDC9, numbered 1-9 in the diagrams).
3.4. Worst case analysis
The reserve ratio (or equivalently the funding ratio) is an indicator of the burden (or the
wealth) one generation hands over to the next generation. It is clear that a supervisory
authority has to step in if the funding ratio (or equivalently the reserve ratio) falls below a
critical value. In this section we examine the temporal development of the reserve ratio. Our
focus will be to check whether the adjustment components of the ALM rules (2) and (3) work
effectively for real market data. Furthermore, we conduct a worst case analysis by calculating
the minimal reserve ratio for all possible starting times. We do this for our reference model
CDC1 ( 0 = 0.2) and for the cold start variant CDC2 ( 0 = 0).
The results are shown in Fig. 9. It transpires that if the CDC plans started with an initial
reserve ratio of 0.2, the reserve never falls substantially below zero. Actually, for all starting
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times after 1980 the minimal reserve ratio remains well above 5%, which indicates that from
that date onwards a more risk seeking ALM-strategy could be have been applied. However, if
the CDC plan were to start without reserves (model CDC2) then for certain starting times the
minimum reserve ratio would fall below zero - cf. Fig. 9. The worst of all possible cases is a
starting time on 1st Aug. 1980. If the CDC-model started at that date with no initial reserve,
then the strict application of the ALM rules would lead to a negative reserve of nearly 10%,
which corresponds to a funding ratio of 90.5%.
Fig. 9. Minimum (log-) reserve ratio depending on the starting time (1st Jan. 1955 earliest, 1st
Sept. 2015 latest) for CDC1 ( 0 = 0.2) and CDC2 ( 0 = 0).
A closer look at this worst case scenario for CDC2 shows that during the first 10 months
(Aug. 1980 until June 1980) the reserve ratio reduces from 0 (starting level) to -9.71% while
at the same time interest rates shot up (money-market rates from 9.16% to 12.94%, yields of
government bonds from 7.80% to 10.80%). This had then two effects on our CDC2-model.
Firstly, REXP-asset lost 7% and secondly, the profit participation went up because of rule (3).
Following rule (3) the profit participation is set to (t) =  (e)(t) +  ((t) - s). The extreme
money market rates which effect  (e)(t) outweigh the stabilizing factor  ( (t) - s).
This observation might be a clue as how to modify rule (3) for a real world implementation of
the CDC-model. However, for our backtesting analysis we stick to the mechanic rule (3)
which was derived from the theoretical model. It should be stressed that neither the dotcom
crisis nor the financial crisis had any substantial impact on the stability of the CDC pension
fund.
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We close this section with the calculation of the risk return profiles in the worst case scenario,
supposing that the CDC plan had started on 1st Aug. 1980. Again we compare all 30-year
saving plans for individual saving vehicles (DAX-, REXP-, MM plans) with the CDC
variants.12
The following Fig. 10a-d corresponds to Fig. 8a-d. Apparently, the risk return profiles in the
worst case scenario do not differ substantially from those in Fig. 8a-d. Only with respect to
the standard deviation the CDC plans are riskier than REXP plans. However, if one looks at
the time span between Aug. 2010 and Sept. 2015, we see that interest rates steadily decreased
and so did the yields at maturity of MM- and REXP plans. This explains the high values for
the standard deviation of MM- and REXP plans in Fig. 8a. The same holds for CDC plans in
that time period. The decreasing yields demonstrate that the CDC plan steadily adjusts to a
changing economic environment of lower interest rates.
12
There are in total 62 saving plans between 01.08.1980 and 30.09.2015.
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Fig. 10a-d. Risk return-profile of 30-year saving plans in the worst case scenario: Mean value
of annualized yield at maturity versus four risk indicator: standard deviation (Fig. 10a, top
left), intergenerational imbalance (Fig. 10b, top right), path volatility (Fig. 10c, bottom left)
and maximum drawdown (Fig. 10d, bottom right). We compare individual saving plans (DAX
- REXP mixed portfolios, MM plan) and with CDC-variants (CDC1-CDC9, numbered 1-9 in
the diagrams).
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4. Conclusion and outlook
Our analysis shows that collective defined contribution (CDC) schemes are more than just a
nice idea. Based on the theoretical model presented in (Goecke, 2013) we could prove that the
risk return profile for CDC plans is much better than that for DC plan. Even in a worst case
scenario CDC plans perform better than individual DC plans. Clearly, there is no guarantee
that an excellent performance observed in the past will recur in future. But it should be
stressed that our analysis is based on observations of 60 years. Within this time span, we have
observed extreme situations such as the oil price crisis 1973, stock market crashes and
bubbles, high inflation rates, and extreme short and long term interest rates.13 The strength of
the CDC plan is that it is self adjusting due to resilience factors in the ALM rules. With
respect to the fundamental objective to ensure a fair participation in production factor capital,
a CDC plan is superior to a pension plan with an interest rate guarantee. Looking back into
the economic history, we should realize that long term interest rates either are worthless or are
unbearable for the warrantor. Pension systems must be adjustable to the economic reality.
It has been criticized that a CDC plan is just a zero sum game: the advantage for one
generation of savers is the disadvantage for the other. This view totally ignores the principle
of risk sharing and insurance. A CDC scheme is an “insurance” contract where the saver pays
a premium (in form of contributions into the collective reserve) and receives benefits (in form
of payments out of the collective reserve). In A Theory of Justice John Rawls introduced the
concept of fairness under the veil of ignorance.14 Following Rawls’ theory a transgenerational
contract between savers will be regarded as fair, if the savers were to agree upon this contract
provided they did not know in advance which generation they belong to – i.e. under the veil of
ignorance. From this perspective a CDC plan is fair. However, it is also clear that a saver
13
14
The 1-month money market rates ranged between 13.33% (Sept. 1973) and – 0.11% (Sept. 2015).
Cf. (Rawls 1991), in particular Section 24 (pp. 118ff.) and Section 44 (pp. 251ff.)
10.02.2016 18:31
30
might feel unfairly treated if she or he was obliged to enter a heavily underfunded CDC
scheme.
CDC schemes are not an all-purpose answer to the pension challenge in an aging society.
There are apparent limitations. For example a CDC plan can only work if the participants
cannot withdraw money at will. Contractual compliance is essential for intergenerational risk
transfer. That does not mean that CDC plans are Ponzi plans. Whenever the flow of new
entrants stops, a CDC plan can be converted into a simple DC plan by setting the target
reserve ratio to zero.
Our backtesting analysis considered only the accumulation phase. We assumed that the
wealth at retirement is paid out and reinvested outside the fund. An apparent extension of the
model is to include a decumulation phase and then analyze the pension risk, e.g. the volatility
of pensions in payment. It would be equally interesting to analyze population dynamics
(growing or shrinking generations, winding off) in a CDC scheme.
We hope at least, that our analysis has added one more argument in favour of CDC plans as a
good alternative to DC and DB plans.
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34
Appendix: Capital market data for Germany 1955 - 2015
German equity performance index (DAX): The DAX started on 30 Dec. 1987 with a base
value of 1000. It is a performance index calculated on the basis of the 30 largest (measured
according the market value of free floating equities) German listed companies.15 End of
month values for the DAX (from 12.1987 onwards) are provided by Deutsche Bundesbank16.
Backward calculations for the DAX are taken from (Stehle et al. 1996)17. It should be noted
that due to the rules of German corporate tax laws for the years 1977 to 1993 (1994 to 2001)
dividend payments included a 36% (30%) corporate income tax credit. Thus, for these years
the DAX slightly underestimates the pre-tax performance of a German blue chip stocks
portfolio.
German fixed income performance index (REXP): The REXP started on 30 Dec. 1987 with a
base value of 100. It is a performance index calculated on the basis synthetic German
Government bonds with remaining terms between 6 month and 10.5 years. 18 End of month
values for the REXP (including a backward calculation from 01.1967 onwards) are provided
by Deutsche Bundesbank.19 To the best knowledge of the author there are no published
backward calculations for months before Jan. 1967. There we calculate a REXP-proxy for the
time period 12.1954 to 12.1966, based on Government bond interest rates reported by
Deutsche Bundesbank.20
Month Money market interest rates m(t) for t: 01.1955 to 09.2015: The 1-month money
market interest rates series is a concatenation of data provided by Deutsche Bundesbank.21
15
For details c.f. (Deutsche Börse AG, 2015a)
Time series BBK01.WU3141
17
Table 3, p. 296
18
For details c.f. (Deutsche Börse AG, 2015b)
19
Time series BBK01.WU046A
20
Time series BBK01.WU0115 (03.1960- 01.1966), BBK01.WU0018 (05.1956- 02.1960) and 6.10% for the 5
months 12.1954 - 04.1955.
21
Time series BBK01.SU0310 (from 01.1999 onwards), BBK01.SU0104 (12.1959- 12.1998) and values taken
from the Statistical Supplements of the Monthly Reports of Deutsche Bundesbank (01.1955 – 12.1958).
16
10.02.2016 18:31
Impressum
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Forschung am IVW Köln, 5/2016
ISSN (online) 2192-8479
Goecke: Collective Defined Contribution Plans – Backtesting based on
German capital market data 1955 – 2015.
Köln, Februar 2016
Schriftleitung / editor’s office:
Prof. Dr. Jürgen Strobel
Herausgeber der Schriftenreihe / Series
Editorship:
Institut für Versicherungswesen /
Institute for Insurance Studies
Prof. Dr. Lutz Reimers-Rawcliffe
Prof. Dr. Peter Schimikowski
Prof. Dr. Jürgen Strobel
Fakultät für Wirtschafts- und Rechtswissenschaften /
Faculty of Business, Economics and Law
Technische Hochschule Köln /
University of Applied Sciences
Gustav Heinemann-Ufer 54
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+49 221 8275-3277
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Institut für Versicherungswesen /
Institute for Insurance Studies
Fakultät für Wirtschafts- und Rechtswissenschaften /
Faculty of Business, Economics and Law
Technische Hochschule Köln /
University of Applied Sciences
Gustav Heinemann-Ufer 54
50968 Köln
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